Hello!
If I login directly as root, then Adblock Plus works perfectly fine with all it's settings working and all it's blocking capabilities.
But if I login as regular user, then Adblock Plus is not blocking anything and it's settings are impossible to change through GUI or otherwise.

Hello!
The contents of my ~/.mozilla directory are all belong to user (victor:victor) and I never use root directly, I use doas when it need to perform as root. However in this particular case when all firefox add-ons work well and Adblock Plus refuses to work, I decided to login directly as root, launch firefox, add Adblock Plus from official mozilla add-ons menu and Adblock Plus worked fine. Then I loged back into my regular user and tried different things like removing Adblock Plus and adding it directly from their website, but nothing works.
I searched on the Internet for this issue, but the only thing I was able to find that another ad blocker Ublock Origin also not working properly under OpenBSD, failing to block ads.
I should mention, I use OpenBSD 6.5

I'd suggest to move ~/.mozilla to ~/.mozilla_old and start firefox from a X terminal.
Reinstall the extensions and if the problem reappears check the output from the terminal to get a clue of what is happening.

You can also check that your home directory isn't full and that you still have writing rights.
I have been using Ublock Origin for a while without any issue.

I checked home directory and .mozilla for writing rights and everything is fine. Then I moved .mozilla to .mozilla_old and started firefox from terminal, as you suggested. Then I added Adblock Plus and it wasn't working. If I click on Adblock Plus icon on the toolbar, nothing happens; if I go to add-ons menu and click "Preferences" next to Adblock Plus, it loads options.html, where I am unable to activate any checkboxes and change any settings. As a result it's not blocking any ads.
Terminal don't output any errors.

You said the matter was not related to the recent Mozilla certificate issue but this problem, AFAIK, was fixed on 60.6.2esr.
When was Adblock Plus working for the last time (if ever) and how did you make your Add-ons work again ?

If you are running Release/Stable you can upgrade Firefox-esr through the ports collection and via packages if you are using Current.

No, the certificate issue was about an expired certificate, which automatically marked all addons as non-compatible.
Mozilla pushed new certificate through it's "studies" feature - it's a file cert9.db, I have that file copied into new profile.
Adblock Plus never worked for me. It is listed in add-ons as installed, it shows in the menu bar, but I can't change any of it's options on a preferences page and therefore it's not blocking anything. I've been using OpenBSD since release of 6.5 (I'm trying to move away from Linux), and the first thing I notice, that Adblock Plus is not working in Firefox, but every other add-on is working fine.

I just tried Adblock Plus on Firefox 66 and it's not working.
uBlock Origin is working fine, thanks.
I think I'll use uBlock Origin as a temporary solution.

EDIT:

I've been thinking about this problem and this is what comes to my mind:
When I click add-ons and then Preferences for Adblock Plus, it opens a file
moz-extension://....../options.html
It's either 1) the regular user don't have permissions to write to options.html or
2) options.html don't have permisions to store it's data somewhere else.
The question is: where exactly options.html is located?
doas find / -name "options.html" returned no results.